PRELIMINARY NOTES ON RIGID REFLECTON GROUPS

H. E. A. CAMPBELL, I. P. HUGHES, AND R. J. SHANK

Introduction

We are interested in the problem of characterising the representations of finite groups with polynomial invariants. It is well known that for a non-modular representation the ring of invariants is a polynomial algebra if and only if the group is generated by reflections. There are, however, modular reflection groups with non-polynomial invariants. In a 1982 paper in the Bulletin of the A.M.S.([4]) Victor Kac conjectured a sufficient condition for a representation to have polynomial invariants. The primary purpose of this note is to give a counter example to this conjecture.

In Section 1 we give the required definitions, state the relevant background materal and formulate the rigidity conjecture. In Section 2 we state Kac’s conjecture and prove that it is equivalent to the rigidity conjecture. Section 3 contains an example illustrating the significance of the fact that Kac’s conjecture was stated for representations over an algebraically closed field. Section 4 contains our counter example.

1. Preliminaries

Let V be a finite dimensional vector space over a field k and denote the symmetric algebra by S(V^{∗}) . If k is algebraically closed then S(V^{∗}) is the set of polynomial functions on V. Choosing a basis for V with dual basis {x_{i}}_{i=1}^{n} gives an isomorphism between S(V^{∗}) and k[x_{1}, …, x_{n}]. Let G be a subgroup of GL(V). There is an induced left action of G on S(V^{∗}). We are interested in the ring of invariant polynomials:

S(V^{∗})^{G} = {f ∈ S(V^{∗}) | ∀g ∈ G(f · g = f)}.

Date: September 16, 2002.

1991 Mathematics Subject Classification. 13A50.

Key words and phrases. Invariant theory.

This research is supported in part by the Natural Sciences and Engineering Research Council of Canada.

If S(V^{∗})^{G} is isomorphic to a polynomial algebra we say that G has polynomial invariants.

An element, r, of GL(V) is a reflection if its order is finite and the rank of r − 1_{G} is one. In the literature, these elements are sometimes called pseudo-reflections. A reflection group is a subgroup of GL(V) generated by reflections.

For the remainder of the paper we assume that G is a finite subgroup of GL(V). We will say that G is modular if the characteristic of k divides the order of G.

Theorem 1.1. Suppose G is non-modular. Then G has polynomial invariants if and only if G is a reflection group.

Proof. See [3], [6] and Ch.5 of [2] .

There are modular reflection groups with non-polynomial invariants. (See Section 3 or Section 4.). In an attempt to characterise the modular reflection groups with polynomial invariants we introduce the concept of a rigid reflection group. If U is a subspace of V then the associated isotropy subgroup is

G_{U} = {g ∈ G | ∀u ∈ U(g · u = u)}.

We say that G is rigid if each isotropy subgroup is a reflection group. By taking U = {0} we see that every rigid group is a reflection group.

Theorem 1.2. If G has polynomial invariants then G is rigid.

Proof. See Ch.5, Sec.5, Ex.8 of [2].

It is a consequence of Theorem 1.2 and Theorem 1.1 that all nonmodular reflection groups are rigid. There are modular reflection groups which are non-rigid (See Section 3). Since many familiar examples of non-rigid groups have non-polynomial invariants, we made the following conjecture.

Rigidity Conjecture: G has polynomial invariants if and only if G is rigid.

Unfortunately, as we shall see in Section 4, this conjecture is false.

2. The Equivalence of Kac’s conjecture and the rigidity conjecture

In this section we state Kac’s conjecture (See [4]) and prove that it is equivalent to the rigidity conjecture.

Kac’s Conjecture: Suppose that k is algebraically closed. Then G has polynomial invariants if and only if for each v ∈ V, G_{v} is a reflection group.

We will say that G satisfies the Kac condition if for every v ∈ V, G_{v} is a reflection group. Let \overline{k} denote the algebraic closure of k and let \overline{G} denote the image of G in GL(V ⊗ \overline{k}).

Theorem 2.1. G is rigid if and only if \overline{G} satisfies the Kac condition.

Proof. We start with a few observations. A field extension preserves the rank of a matrix; therefore g ∈ G is a reflection if and only if the corresponding \overline{g} ∈ \overline{G} is also a reflection. Suppose that H is a subgroup of G and \overline{H} is the image of H in \overline{G}; then H is a reflection group if and only if \overline{H} is a reflection group. For any g ∈ G, ker(\overline{g}−1) = ker(g−1)⊗\overline{k}; therefore for any subspace U ⊆ V we have \overline{G_{U}} = \overline{G}_{U⊗\overline{k}}.

Suppose that G is rigid. Choose an arbitrary v ∈ V ⊗ \overline{k} and let

U(v) = ⋂_{\overline{g}∈\overline{G}_{v}} ker(g − 1).

U(v) is a subspace of V. G is rigid so G_{U(v)} and \overline{G_{U(v)}} are reflection groups. By definition \overline{G}_{v} ⊆ \overline{G_{U(v)}}. Since ker(\overline{g} − 1) = ker(g − 1) ⊗ \overline{k} we have v ∈ U(v) ⊗ \overline{k} and \overline{G_{U(v)}} ⊆ \overline{G}_{v}. Thus \overline{G}_{v} = \overline{G_{U(v)}} and \overline{G}_{v} is a reflection group.

Suppose \overline{G} satisfies the Kac condition. Choose an arbitrary subspace U ⊆ V. We need to show that \overline{G_{U}} = \overline{G}_{U⊗\overline{k}} is a reflection group. To do this we will find an element v ∈ U ⊗ \overline{k} such that \overline{G}_{v} = \overline{G}_{U⊗\overline{k}}.

If \overline{G}_{U⊗\overline{k}} = \overline{G} then take v to be zero. Otherwise, let \{g_{i}\}_{i=1}^{m} represent the distinct cosets of G_{U} in G. Let U_{ij} = ker(g_{i}^{−1}g_{j} − 1) ∩ U and Ũ = ∪_{i≠j}U_{ij} ⊗ \overline{k}. Ũ is a union of proper subspaces of U ⊗ \overline{k} and \overline{k} is infinite; therefore Ũ ≠ U ⊗ \overline{k}. Choose v ∈ U ⊗ \overline{k} − Ũ. Since v ∈ U ⊗ \overline{k} we have \overline{G}_{U⊗\overline{k}} ⊆ \overline{G}_{v} and the orbit of v contains at most m elements. However, when i ≠ j, v ∉ U_{ij}; therefore g_{i}v ≠ g_{j}v and the orbit of v contains at least m elements. Thus the orbit of v contains exactly m elements and \overline{G}_{v} = \overline{G}_{U⊗\overline{k}}.

Since G has polynomial invariants if and only \overline{G} has polynomial invariants, it follows from Theorem 2.1 that the rigidity conjecture is equivalent to Kac’s conjecture. In Section 4 we give a counter example to the rigidity conjecture.

3. An Example: F_{4} at p = 3

In this section we give an example of a non-rigid representation which satisfies the Kac condition. Since the representaion is over the finite field F_{3}, this is consistant with Theorem 2.1.

Let K be a compact Lie group with Weyl group W (K). Let V denote the elements of order p in a maximal torus, T. We think of V as a vector space over F_{p}. The action of W (K) on V is then linear giving a representation over F_{p}. We will call this representation the mod-p reduction of the Weyl group.

Theorem 3.1. If K is a connected, simply connected compact Lie group then the mod-p reduction of W (K) satisfies the Kac condition.

Proof. See for example 2.15 of [7], Sec. 3 of [1] or Prop. 15 of [5].

We now give a second description of the mod-p reduction of W (K). Let h denote the Cartan subalgebra of Lie algebra of K. h is the universal cover of T. The kernel of the projection from h to T, which we denote by Λ, is called the integral lattice. The action of W (K) on T lifts to h and restricts to Λ. For any λ ∈ Λ, exp(λ/p) is an element of V. This assignment induces an W (K)-module isomorphism between Λ ⊗ F_{p} and V. A root system for K can be used to give an explicit description of the action W (K) on h and, therefore, on Λ ⊗ F_{p}. Note that the generating reflections associated to the root system are still reflections over Λ ⊗ F_{p} and thus the mod-p reduction of W (K) is a reflection group.

For any simple compact Lie group the highest coroot can be written as an integral linear combination of the simple coroots. If a prime number, p, is one of the coefficients then there is p-torsion in the cohomology of the Lie group (See 1.12, 2.2 and 2.5 of [7]) and, usually, the mod-p reduction of the Weyl group will have non-polynomial invariants.

Theorem 3.2. The mod-3 reduction of W (F_{4}) is not rigid.

Proof. We prove the theorem by finding a subspace of Λ ⊗ F_{3} whose isotropy subgroup is not a reflection group. We use the usual root system for F_{4}. See, for example, [2].

The second coefficient of the highest coroot is 3. Let v be the element of V corresponding to the second fundamental co-weight ω_{2}. Following 2.14 and 2.15 of [7] or Prop. 15 of [5], the isotropy subgroup, W (F_{4})_{v}, is generated by reflections corresponding to the roots which vanish on v. This subroot system is of type A_{2} × A_{2} with simple roots {α_{0}, α_{1}, α_{3}, α_{4}}. Here α_{0} is −1 times the highest root. Note that because the second coefficient of the highest coroot is 3, r_{0} fixes ω_{2} in Λ⊗F_{3}. Let K be the centralizer of v in F_{4}. Choose a Coxeter element, g = r_{0}r_{1}r_{3}r_{4}, for W (K). Note that |g| = 3. Working over Λ ⊗ F_{3}, the only eigenvalue for g is 1. The eigenspace, U, has dimension 2 with a basis given by {ω_{2}, ω_{1} −ω_{3} −ω_{4}}. Since the rank of g −1 is 2, g is not a reflection and < g > is not a reflection group. We complete the proof by showing that W (F_{4})_{U} =< g >.

Let t be the element of T corresponding to ω_{1} − ω_{3} − ω_{4}. We apply 2.14 and 2.16 of [7] to K. No root of K vanishes on t. Thus W (K)_{t} is isomorphic to a subgroup of the fundamental group of K. Note that g ∈ W (K)_{t}. K is of type A_{2} × A_{2} so Z/3 × Z/3 is an extension of the center of K by the fundamental group of K. Since < v > is a subgroup of the center and < g > is isomorphic to a subgroup of the fundamental group, the fundamental group is isomorphic to < g >. Therefore W (K)_{t} =< g >. Since W (F_{4})_{U} = W (K)_{t}, the result follows.

Note that F_{4} satisfies the hypotheses of Theorem 3.1 and therefore the mod-3 reduction of W (F_{4}) satisfies the Kac condition. Also, it follows from Theorem 1.2 that the ring of invariants for the mod-3 reduction of W (F_{4}) is not a polynomial algebra.

4. A Counter Example

In this section we provide an example of representation of (Z/p)^{3} which is rigid and has non-polynomial invariants. This gives a counter example to the rigidity conjecture and, in light of Section 2, a counter example to Kac’s conjecture.

Consider

G = \{( \begin{matrix}1 & 0 & a & c \\ 0 & 1 & c & b \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} ) | a, b, c ∈ F_{p}\} .

G is a subgroup of GL_{4}(F_{p}) isomorphic to (Z/p)^{3}. We specify elements of G by giving the upper right 2 × 2 submatrix. G is generated by the following set of reflections:

\{r_{1} = ( \begin{matrix}1 & 0 \\ 0 & 0 \end{matrix} ) , r_{2} = ( \begin{matrix}0 & 0 \\ 0 & 1 \end{matrix} ) , r_{3} = ( \begin{matrix}1 & 1 \\ 1 & 1 \end{matrix} )\} .

Theorem 4.1. G is rigid.

Proof. We start by describing the isotropy subgroups of the one dimensional subspaces. Choose v ∈ V and let c_{i} denote the coefficients of v with respect to the standard basis for the column space.

(i) If c_{3} = c_{4} = 0 then G_{v} = G.

(ii) If c_{3} = 0 and c_{4} ≠ 0 then G_{v} is generated by r_{1}.

(iii) If c_{3} ≠ 0 and c_{4} = 0 then G_{v} is generated by r_{2}.

(iv) If c_{3} ≠ 0 and c_{4} ≠ 0 then G_{v} is generated by ( \begin{matrix}c_{4}/c_{3} & −1 \\ −1 & c_{3}/c_{4} \end{matrix} ).

Each of these subgroups is a reflection group. The isotropy subgroups for the higher dimensional subspaces are formed by taking intersections. The only new subgroup formed from these intersections is the trivial subgroup. Thus all of the isotropy subgroups are reflection groups and G is rigid.

Remark 4.2. Since each proper non-trivial isotropy subgroup is generated by a single transvection, the ring of invariants for each of these groups is a polynomial algebra.

Theorem 4.3. S(V^{∗})^{G} is not a polynomial algebra.

Proof. If S(V^{∗})^{G} is a polynomial algebra it has four generators and the product of the degrees of the generators is |G| = p^{3}. Observe that x_{3} and x_{4} are invariant and that the dimension of S(V^{∗})^{G} in degree one is 2 (We remind the reader that elements of V^{∗} should be treated as row vectors.). Thus if S(V^{∗})^{G} is a polynomial algebra it has one generator in degree p and one in degree p^{2}. We will show that S(V^{∗})^{G} does not have a generator in degree p.

Consider the subgroup of G given by

H = \{( \begin{matrix}a & 0 \\ 0 & b \end{matrix} ) | a, b ∈ F_{p}\} .

H is isomorphic to (Z/p)^{2}. Let y = x_{1}^{p} − x_{1}x_{3}^{p−1} and z = x_{2}^{p} − x_{2}x_{4}^{p−1}. Both y and z are H-invariant elements of degree p and, therefore, S(V^{∗})^{H} ≅ F_{p}[y, z, x_{3}, x_{4}]. Since S(V^{∗})^{G} ⊆ S(V^{∗})^{H}, the degree p elements of S(V^{∗})^{G} are contained in the vector space spanned by y, z

and monomials in x_{3} and x_{4}. Observe that ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) does not fix any

non-trivial linear combination of y and z. Hence S(V^{∗})^{G} does not have a generator in degree p and is not a polynomial algebra.

Remark 4.4. We believe that the ring of invariants for the above example is a hypersurface with two generators in degree 1, one generator in degree p + 1, two generators in degree p^{2}, and a relation in degree (p + 1)p.

Note that our counter-example is a p-group. The reflections contained in a p-group are not diagonalizable even after extending the field. With this in mind we suggest the following.

Modified Rigidity Conjecture: If G is rigid and generated by diagonalizable reflections then G has polynomial invariants.

References

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2. N. Bourbaki, Groupes et algèbres de Lie, Ch. 4,5 et 6 , Éléments de Mathématique, Masson, Paris, 1981(1968).

3. C. Chevalley, Invariants of finite groups generated by reflections, Amer. J. Math. 67 (1955), 778–782.

4. V. Kac and K. Watanabe, Finite linear groups whose ring of invariants is a complete intersection, Bull. A.M.S. 6 (1982), no. 2, 221–223.

5. A. Pianzola, On the arithmetic of the representation ring and elements of finite order in Lie groups, J. Algebra 108 (1987), 1–33.

6. G. C. Shephard and J. A. Todd, Finite unitary reflection groups, Can. J. Math. 6 (1954), 274–304.

7. R. Steinberg, Torsion in reductive groups, Adv. in Math. 15 (1975), 63–92.

Department of Mathematics and Statistics, Queen’s University,

Kingston, Ontario, Canada, K7L 3N6

  E-mail address: eddy@mast.queensu.ca, hughesi@queensu.ca, shank@mast.queensu.ca